Coherence and Control of Quantum Registers Based on Electronic Spin in a Nuclear Spin Bath

We consider a protocol for the control of few-qubit registers comprising one electronic spin embedded in a nuclear spin bath. We show how to isolate a few proximal nuclear spins from the rest of the bath and use them as building blocks for a potentially scalable quantum information processor. We describe how coherent control techniques based on magnetic resonance methods can be adapted to these solid-state spin systems, to provide not only efficient, high fidelity manipulation but also decoupling from the spin bath. As an example, we analyze feasible performances and practical limitations in the realistic setting of nitrogen-vacancy centers in diamond.

Figure 1

(a) System model, showing the electronic spin in red and surrounding nuclear spins. The closest nuclear spins are used as qubits. Of the bath spins, only the ones outside the frozen core evolve due to dipolar interaction, causing decoherence. (b) Frequency selective pulses, in a 3-qubit register.

Figure 2

Circuits for controlled gates $U$ among two nuclear spins (a) and a nuclear and the electronic spin (b). The gates $A$, $B$, $C$ are defined such that $U=e^{i\alpha }AZBZC$ and $ABC=\mathbb{1}$, where $Z$ is a $\pi$ rotation around $z$ [35]. ${\Phi }_{\alpha }$ is a phase gate: $|0⟩⟨0|+|1⟩⟨1|e^{i\alpha /2}$ and the gate $\alpha$ indicates $e^{i\alpha /2}\mathbb{1}$.

Figure 3

rf pulse scheme for 1 nuclear spin gate in the $m_s=1$ manifold. With $t_p=\beta /\overline{\Omega }$ and fixing ${\psi }_1=\alpha -\lambda$ and ${\psi }_2=\pi -\lambda$, the time delays are $t_1=\frac{T}{2}-t_p-t_{\pi }-\frac{\alpha +\gamma }{\omega }$ and $t_2=\frac{T}{2}-t_{\pi }+\frac{\alpha +\gamma }{\omega }$. This yields a minimum clock time $T\ge 4\pi {\mathrm{Max}}_{{\overline{\Omega }}_j,{\omega }_1^j}\{{\overline{\Omega }}_j^{-1}+({\omega }_1^j{)}^{-1}\}$.

Figure 4

rf and $\mu w$ pulse sequence to implement a 1 nuclear spin gate while reducing the effects of a slowly varying electronic dephasing noise. The black bars indicate $\pi$-pulses, while the first rectangle indicate a general pulse around the $x$ axis. $\tau =T/8-t_{\pi }/2$ and $t_{\phi }=(\alpha +\gamma )/4\omega$.